A326692 Values of k for which the denominator of (Sum_{prime p | k} 1/p - 1/k) is k.

1, 4, 8, 9, 15, 16, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 49, 51, 52, 60, 63, 64, 65, 68, 69, 72, 76, 77, 81, 85, 87, 88, 91, 92, 95, 96, 99, 100, 104, 108, 112, 115, 116, 117, 119, 121, 123, 124, 125, 128, 133, 135, 136, 140, 141, 143, 144, 145, 148

Offset: 1

Author

Jonathan Sondow, Jul 20 2019

Keywords

• nonn

Comments

Any prime power p^k with k > 1 is a term, as 1/p - 1/p^k = (p^(k-1) - 1)/p^k which is in reduced form and has denominator p^k.

Are there infinitely many Carmichael numbers A002997 in the sequence?

Examples

			1/3 + 1/5 - 1/15 = 7/15 has denominator 15, so 15 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002997, A326689, A326690, A326691, A326715.

Programs

Mathematica

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == # &]

PARI

is(k) = {my(p = factor(k)[,1]); denominator(sum(i = 1, #p, 1/p[i]) - 1/k) == k;} \\ Amiram Eldar, Apr 26 2024

Formula

Solutions of A326690(x) = x. That is, fixed points of A326690.